課程名稱︰微積分甲下 課程性質︰必修 課程教師︰薛克民 開課學院：電資學院、工學院、管理學院 開課系所︰電機系、資工系、材料系、資管系 考試日期（年月日）︰100/4/11 考試時限（分鐘）：40分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : ˙每題十分 ˙請詳述計算過程，無計算過程的答案不予計分 1. Find the tangent, normal and curvature of the curve γ(t) = ( cos(t) + t*sin(t) , sin(t) - t*cos(t) , t ). (10%) 2. At the point (1,3), a function f(x,y) has a derivative of -√5 in the direction from (1,3) toward (2,1) and a derivative of √5 in the direction from (1,3) toward (3,2). Find the directions in which the function f increases most rapidly at (1,3), and find the rates of change in this direction. (10%) 3. Determine whether the following limits exist. If it does, find the value. (10%) 5x*√y (a) lim ──────── (x,y)→(0,0) √(x^3 + y^3) 2 2 (x^2 + y^2) x y e (b) lim ───────── (x,y)→(0,0) x^4 + y^4 4. Let u = u(x,y), where x = x(s,t) and y = y(s,t), and assume that all these functions have continuous second derivatives. show that 2 2 2 2 2 2 δ u δ u ┌ δx ┐ δ u δx δy δ u ┌ δy ┐ ─── = ───│ ──│ + 2 * ─── * ──* ── + ───│ ──│ δs^2 δx^2 └ δs ┘ δxδy δs δs δy^2 └ δs ┘ 2 2 δu δ x δu δ y + ──* ─── + ──* ─── δx δs^2 δy δs^2 --

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